Do you have an external DC PSU that you can power the phone from? If you do you can measure the current draw of the phone.
Also, if you have a Mac, search for a program called "coconut battery" it will help you diagnose whether your battery is in need of replacing.
I have also found that more often than not, when you replace a battery on a 5S, you also have to restore the phone - and no back-ups allowed.
Those are some things to consider before you just dive into the motherboard